When students are taught about ratios and unit rates together, they often mix up using a unit rate, and a simplified ratio. For example, when asked to write a simplified ratio for "5 boys for every 10 girls", a student who recently learned about unit rates may use a calculator and write 0.5 boys/1 girl. This is not an incorrect statement, but does not reflect understanding of the difference between a unit rate and a reduced ratio (1 boy / 2 girls). How can a teacher address this misunderstanding?
This activity correlates with Utah Pre-Algebra Curriculum Standard 2, Objective 1 (Model and illustrate meanings of ratios, percents, and decimals), Letter b (Compare ratios using the unit rate). It also corresponds with Math 7 Objective 1, Standard 1, Letter d (Select the most appropriate form of a rational number for a given context)
Things To Understand:
According to Math.Com "A rate is a ratio that compares two different kinds of numbers, such as miles per hour or dollars per pound. A unit rate compares a quantity to its unit of measure." The key to this is that a unit rate compares to the "unit of measure". For example, a person does not usually say "I am traveling 100 miles / 2 hours." The reason we don't is that we don't measure the unit of 2 hours. The unit of measure is 1 hour, so we say, "I am traveling 50 miles / (1) hour." It becomes apparent that the denominator, will always be one. The reason we do this is to have a uniform measurement by which we can compare things.
Possible Student Challenges or Misunderstandings:
The student does not see the unit rate as a standardized measurement.
$150 for 9 hours of work = $16.67 per hour
The student does not understand the reason, or value of a unit rate.
$150 for 9 hours of work = $16.67 per hour = Better Rate
$65 for 5 hours of work = $13 per hour = Lower Rate
One colleague suggested that students would better understand slope as a measure of steepness if we had students express it as a unit rate.
The student does not understand the reason or value of a ratio.
Unit rates compare different types of data.
Ratios for part to whole relationships.
3 out of 5 boys makes more sense than 0.6 boys per 1 boy.
45 out of 50 points-not .9 points for every point.
Addressing the Misunderstanding:
Is this a student problem, or a teacher problem?
Is the student expressing a comparison in a way that makes more sense than what the teacher expects?
If you wanted to compare which class has a higher "concentration" of boys 0.5 boys per girl or 0.6 boys per girl might make sense.
Are examples used for unit rates, and ratios that don't allow students to make real life connections?
3.5 books per pencil would possibly not be as a good of a unit rate as would 35 pages per hour. Students are more likely to experience measuring pp./minute as opposed to books/pencil.
Following the reflection in step 1, choose some possible uses of ratios and unit rates that make sense.
Create a need for students to use ratios over unit rates, and unit rates over ratios. When a student encounters a situation that using a ratio is not useful, or using the unit rate is not useful, they will be more likely to understand the best times to use one or the other.
Unit Rates over Ratios: Using the US Government Quick Facts page, gather some demographic information about states square mileage and population. List population and land area for 5-10 of the states and ask students to order them from least crowded to most crowded.
Ratios over Unit Rates: List 5 or 10 test scores as unit rates (ie. 0.9 points obtained/1point possible). Tell students the test was worth 40 points, ask them to figure out the score for each test. Ask them how the 0.9 points per point is useful. Ask them how the actual score is more useful.
Assessment: As a homework assignment ask students to locate examples of unit rates in real life, and examples of ratios in real life, and to explain why they think the ratio, percent, or unit rate was expressed in the format provided. By the examples they provide, and by their explanations it should become apparent who recognizes the appropriate times to use which. As part of the assignment you can provide them with relationships that are best represented by ratios, and some best represented by unit rates. Ask students to justify their reasoning. If their justification is sound, and the representation they choose is useful it will be a strong indicator of understanding.
When students are taught about ratios and unit rates together, they often mix up using a unit rate, and a simplified ratio. For example, when asked to write a simplified ratio for "5 boys for every 10 girls", a student who recently learned about unit rates may use a calculator and write 0.5 boys/1 girl. This is not an incorrect statement, but does not reflect understanding of the difference between a unit rate and a reduced ratio (1 boy / 2 girls). How can a teacher address this misunderstanding?
This activity correlates with Utah Pre-Algebra Curriculum Standard 2, Objective 1 (Model and illustrate meanings of ratios, percents, and decimals), Letter b (Compare ratios using the unit rate). It also corresponds with Math 7 Objective 1, Standard 1, Letter d (Select the most appropriate form of a rational number for a given context)
Things To Understand:
According to Math.Com "A rate is a ratio that compares two different kinds of numbers, such as miles per hour or dollars per pound. A unit rate compares a quantity to its unit of measure." The key to this is that a unit rate compares to the "unit of measure". For example, a person does not usually say "I am traveling 100 miles / 2 hours." The reason we don't is that we don't measure the unit of 2 hours. The unit of measure is 1 hour, so we say, "I am traveling 50 miles / (1) hour." It becomes apparent that the denominator, will always be one. The reason we do this is to have a uniform measurement by which we can compare things.
Possible Student Challenges or Misunderstandings:
Addressing the Misunderstanding: